On complete and incomplete sets of observables, the principle of maximum entropy - Revisited

R. S. Ingarden is one of the founders of the principle of maximum entropy ( formulated independently by E. T. Jaynes), an important approach to basic a ssumptions of statistical physics. He is primarily responsible for recogniz ing this principle and fostering its development (cf. e.g, [1-3]). In this paper we discuss some questions connected, in natural way, with the princip le of maximum entropy, namely, the question of the minimal number eta of ob servables Q(1)....,Q(eta), whose mean values at some instants t(1),...,t(s) determine the statistical state of an N-level open quantum system (we assu me that the time evolution of the system in question is given by a semigrou p of linear transformations with a generator L) and a similar problem of th e minimal number kappa of instants t(1),...,t(kappa), which are necessary f or reconstructibility of trajectories of the system. To give an exact answe r to the above questions we introduce the following two concepts: index of cyclicity of a given generator L of time evolution and index of reconstruct ibility (for a given generator L and a given set of observables Q(1),...,Q( n)).